Determining the magnitude of a certain frequency on the iPhone

Question

I'm wondering what's the easiest/best way to determine the magnitude of a given frequency in a sound.

It's my understanding that a FFT function will return the magnitudes of all frequencies in a signal. I'm wondering if there is any shortcut I could use if I'm only concerned about a specific frequency.

I'll be using the iPhone mic to record the audio. My guess is that I'll be using the Audio Queue Services for recording since I don't need to record the audio to a file. I'm using SDK 4.0, so I can use any of the functions defined in the Accelerate framework (e.g. FFT functions) if needed.

Update: I updated the question to be more clear as per Conrad's suggestion.

Answer 1

Your first problem is defining your problem more definitively, because a "sound" doesn't have "a" frequency; in general it contains many frequencies.

So you probably want the FFT, but you'll have to define your own rules for which of the many frequencies the FFT returns represents "the" frequency you want to test.

If we knew more about the real-world problem you're trying to solve, that might help.

Answer 2

If you only need to test for one frequency, you can just calculate the corresponding point of the DFT. The DFT algorithm is O(N^2), but the FFT algorithm reuses intermediate results to achieve O(NlogN) for computation of the DFT. However, if you want only one frequency sample, you can just calculate one output sample of the DFT and achieve O(N) performance.

This can be done by looking at the equation for the DFT on the wikipedia page (I'm not even going to try to type it here) and just calculate Xk for a single k corresponding to the frequency of interest. k is just the indexing on the output of the DFT.

Mapping k (indexes of the DFT output) into real frequencies (Hz) depends on two things:

• Sampling frequency (for example, 44100 Hz for CD Audio)
• FFT size

Real frequencies are mapped to k as follows:

F = k*Fs/N  for k = 0 ... N/2-1 ((N-1)/2 for odd N)

or

k = F*N/Fs  for F = 0Hz ... Fs/2-Fs/N

where F is the frequency in Hz, N is the FFT size, and Fs is the sampling frequency (Hz). Some things to note:

• k is an integer, so not all frequencies will map to an integer k. Find the closest k
• If you need more frequency resolution, increase N.
• Signals sampled at Fs are only able to accurately represent frequencies up to, but not including Fs/2 (Nyquist rate). This is why I showed that the mapping from k to Hz is only good for half the output samples. I will not go into what the second half represents (it will actually be a mirror image of the first half for a real input signal)
• The output of the DFT/FFT is complex. You most likely want to take the magnitude of this.
• If you need to compute even a few DFT outputs, it may be better to just use the FFT function available and get all the output samples instead of calculating just the output samples you need using the DFT. The reason is that most FFT algorithms are heavily optimized so even though you may be theoretically doing less work, it may take longer than the FFT. You would probably just have to benchmark this to see which approach is better.

I've left out quite a few other details for simplicity's sake that shouldn't matter for your application